← Find right and wrong counts from total score · Two-Category Counts from a Total

Find right and wrong counts from total score · 10 practice problems

3.OA.D.83.OA.A.3

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Generated variants — 10

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Variant 1 easy

In a math quiz contest, you earn $3$ points for each question you get right and lose $2$ point for each question you get wrong. If Leo solved $30$ questions in this contest and ended up with $65$ points, how many questions did Leo get wrong?

Your answer

Show solution

1 · Understandwhat's really being asked

Leo answered 30 quiz questions. Each correct answer is worth +3 points and each wrong answer is -2 point. The final score was 65 points. We must find how many questions were wrong.

Givens

Total questions answered: 30
Right answer: +3 points
Wrong answer: -2 point
Final score: 65 points

Unknowns

The number of questions Leo got wrong

Constraints

Right count plus wrong count equals 30
Both counts are whole numbers from 0 to 30

2 · Planchoose the strategy

#6 Guess and Check · also uses: #9 Solve an Easier Related Problem#11 Work Backwards

Start from the easier 'all correct' case to get a clean total, then see how each wrong answer changes the score by a fixed amount. The fixed drop per wrong answer lets us divide to find the count.

3 · Execute4 carry out the plan

1Score if every answer were right

#9 Solve an Easier Related Problem 3.OA.A.3

If all 30 answers were correct, Leo would earn 3 points each.

30 \times 3 = 90

Pretending everything is right gives a simple multiplication to anchor the count.

2Find the points lost per wrong answer

#6 Guess and Check 3.OA.B.5

Turning one right answer into a wrong one loses the +3 it would have earned and adds a -2 penalty, so the score drops by 5 for each wrong answer.

3 + 2 = 5

Each mistake costs the missed 3 points plus a 2-point penalty, a steady 5 each time.

3Find how many points were lost in all

#11 Work Backwards 3.OA.D.8

The actual score is 65, which is 25 below the perfect 90.

90 - 65 = 25

The gap from a perfect score is the total damage all the wrong answers caused.

4Divide to find the wrong count

#6 Guess and Check 3.OA.A.3

Each wrong answer costs 5 points, and the total loss is 25, so divide to get the number of wrong answers.

25 \div 5 = 5

Sharing the 25 lost points into groups of 5 tells how many mistakes there were.

Answer: 5 questions

4 · Reviewdoes it hold up?

If 5 are wrong, 25 are right: 25 times 3 = 75, minus 5 = 65, which matches the final score exactly. The counts 25 and 5 add to 30.

Another way: Convert to a single-unknown equation (tool 13): with w wrong, right is 30 - w, so 3 times (30 - w) minus w = 65, giving 90 - 5w = 65 and w = 5.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the perfect score and dividing the lost points into groups of 5
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Reasoning that each wrong answer changes the score by a fixed 5 points
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the perfect score and the score gap across multiple steps

💡Takeaway. Pretend every answer was right, then count how far the score fell; each mistake costs the same amount, so just divide!

Variant 2 easy

In a math quiz contest, you earn $5$ points for each question you get right and lose $3$ point for each question you get wrong. If Ela solved $20$ questions in this contest and ended up with $76$ points, how many questions did Ela get wrong?

Your answer

Show solution

1 · Understandwhat's really being asked

Ela answered 20 quiz questions. Each correct answer is worth +5 points and each wrong answer is -3 point. The final score was 76 points. We must find how many questions were wrong.

Givens

Total questions answered: 20
Right answer: +5 points
Wrong answer: -3 point
Final score: 76 points

Unknowns

The number of questions Ela got wrong

Constraints

Right count plus wrong count equals 20
Both counts are whole numbers from 0 to 20

2 · Planchoose the strategy

#6 Guess and Check · also uses: #9 Solve an Easier Related Problem#11 Work Backwards

Start from the easier 'all correct' case to get a clean total, then see how each wrong answer changes the score by a fixed amount. The fixed drop per wrong answer lets us divide to find the count.

3 · Execute4 carry out the plan

1Score if every answer were right

#9 Solve an Easier Related Problem 3.OA.A.3

If all 20 answers were correct, Ela would earn 5 points each.

20 \times 5 = 100

Pretending everything is right gives a simple multiplication to anchor the count.

2Find the points lost per wrong answer

#6 Guess and Check 3.OA.B.5

Turning one right answer into a wrong one loses the +5 it would have earned and adds a -3 penalty, so the score drops by 8 for each wrong answer.

5 + 3 = 8

Each mistake costs the missed 5 points plus a 3-point penalty, a steady 8 each time.

3Find how many points were lost in all

#11 Work Backwards 3.OA.D.8

The actual score is 76, which is 24 below the perfect 100.

100 - 76 = 24

The gap from a perfect score is the total damage all the wrong answers caused.

4Divide to find the wrong count

#6 Guess and Check 3.OA.A.3

Each wrong answer costs 8 points, and the total loss is 24, so divide to get the number of wrong answers.

24 \div 8 = 3

Sharing the 24 lost points into groups of 8 tells how many mistakes there were.

Answer: 3 questions

4 · Reviewdoes it hold up?

If 3 are wrong, 17 are right: 17 times 5 = 85, minus 3 = 76, which matches the final score exactly. The counts 17 and 3 add to 20.

Another way: Convert to a single-unknown equation (tool 13): with w wrong, right is 20 - w, so 5 times (20 - w) minus w = 76, giving 100 - 8w = 76 and w = 3.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the perfect score and dividing the lost points into groups of 8
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Reasoning that each wrong answer changes the score by a fixed 8 points
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the perfect score and the score gap across multiple steps

💡Takeaway. Pretend every answer was right, then count how far the score fell; each mistake costs the same amount, so just divide!

Variant 3 easy

In a math quiz contest, you earn $4$ points for each question you get right and lose $1$ point for each question you get wrong. If Zoe solved $25$ questions in this contest and ended up with $80$ points, how many questions did Zoe get wrong?

Your answer

Show solution

1 · Understandwhat's really being asked

Zoe answered 25 quiz questions. Each correct answer is worth +4 points and each wrong answer is -1 point. The final score was 80 points. We must find how many questions were wrong.

Givens

Total questions answered: 25
Right answer: +4 points
Wrong answer: -1 point
Final score: 80 points

Unknowns

The number of questions Zoe got wrong

Constraints

Right count plus wrong count equals 25
Both counts are whole numbers from 0 to 25

2 · Planchoose the strategy

#6 Guess and Check · also uses: #9 Solve an Easier Related Problem#11 Work Backwards

Start from the easier 'all correct' case to get a clean total, then see how each wrong answer changes the score by a fixed amount. The fixed drop per wrong answer lets us divide to find the count.

3 · Execute4 carry out the plan

1Score if every answer were right

#9 Solve an Easier Related Problem 3.OA.A.3

If all 25 answers were correct, Zoe would earn 4 points each.

25 \times 4 = 100

Pretending everything is right gives a simple multiplication to anchor the count.

2Find the points lost per wrong answer

#6 Guess and Check 3.OA.B.5

Turning one right answer into a wrong one loses the +4 it would have earned and adds a -1 penalty, so the score drops by 5 for each wrong answer.

4 + 1 = 5

Each mistake costs the missed 4 points plus a 1-point penalty, a steady 5 each time.

3Find how many points were lost in all

#11 Work Backwards 3.OA.D.8

The actual score is 80, which is 20 below the perfect 100.

100 - 80 = 20

The gap from a perfect score is the total damage all the wrong answers caused.

4Divide to find the wrong count

#6 Guess and Check 3.OA.A.3

Each wrong answer costs 5 points, and the total loss is 20, so divide to get the number of wrong answers.

20 \div 5 = 4

Sharing the 20 lost points into groups of 5 tells how many mistakes there were.

Answer: 4 questions

4 · Reviewdoes it hold up?

If 4 are wrong, 21 are right: 21 times 4 = 84, minus 4 = 80, which matches the final score exactly. The counts 21 and 4 add to 25.

Another way: Convert to a single-unknown equation (tool 13): with w wrong, right is 25 - w, so 4 times (25 - w) minus w = 80, giving 100 - 5w = 80 and w = 4.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the perfect score and dividing the lost points into groups of 5
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Reasoning that each wrong answer changes the score by a fixed 5 points
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the perfect score and the score gap across multiple steps

💡Takeaway. Pretend every answer was right, then count how far the score fell; each mistake costs the same amount, so just divide!

Variant 4 medium

In a math quiz contest, you earn $3$ points for each question you get right and lose $1$ point for each question you get wrong. If Ray solved $36$ questions in this contest and ended up with $96$ points, how many questions did Ray get wrong?

Your answer

Show solution

1 · Understandwhat's really being asked

Ray answered 36 quiz questions. Each correct answer is worth +3 points and each wrong answer is -1 point. The final score was 96 points. We must find how many questions were wrong.

Givens

Total questions answered: 36
Right answer: +3 points
Wrong answer: -1 point
Final score: 96 points

Unknowns

The number of questions Ray got wrong

Constraints

Right count plus wrong count equals 36
Both counts are whole numbers from 0 to 36

2 · Planchoose the strategy

#6 Guess and Check · also uses: #9 Solve an Easier Related Problem#11 Work Backwards

Start from the easier 'all correct' case to get a clean total, then see how each wrong answer changes the score by a fixed amount. The fixed drop per wrong answer lets us divide to find the count.

3 · Execute4 carry out the plan

1Score if every answer were right

#9 Solve an Easier Related Problem 3.OA.A.3

If all 36 answers were correct, Ray would earn 3 points each.

36 \times 3 = 108

Pretending everything is right gives a simple multiplication to anchor the count.

2Find the points lost per wrong answer

#6 Guess and Check 3.OA.B.5

Turning one right answer into a wrong one loses the +3 it would have earned and adds a -1 penalty, so the score drops by 4 for each wrong answer.

3 + 1 = 4

Each mistake costs the missed 3 points plus a 1-point penalty, a steady 4 each time.

3Find how many points were lost in all

#11 Work Backwards 3.OA.D.8

The actual score is 96, which is 12 below the perfect 108.

108 - 96 = 12

The gap from a perfect score is the total damage all the wrong answers caused.

4Divide to find the wrong count

#6 Guess and Check 3.OA.A.3

Each wrong answer costs 4 points, and the total loss is 12, so divide to get the number of wrong answers.

12 \div 4 = 3

Sharing the 12 lost points into groups of 4 tells how many mistakes there were.

Answer: 3 questions

4 · Reviewdoes it hold up?

If 3 are wrong, 33 are right: 33 times 3 = 99, minus 3 = 96, which matches the final score exactly. The counts 33 and 3 add to 36.

Another way: Convert to a single-unknown equation (tool 13): with w wrong, right is 36 - w, so 3 times (36 - w) minus w = 96, giving 108 - 4w = 96 and w = 3.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the perfect score and dividing the lost points into groups of 4
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Reasoning that each wrong answer changes the score by a fixed 4 points
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the perfect score and the score gap across multiple steps

💡Takeaway. Pretend every answer was right, then count how far the score fell; each mistake costs the same amount, so just divide!

Variant 5 medium

In a math quiz contest, you earn $4$ points for each question you get right and lose $1$ point for each question you get wrong. If Ben solved $50$ questions in this contest and ended up with $135$ points, how many questions did Ben get wrong?

Your answer

Show solution

1 · Understandwhat's really being asked

Ben answered 50 quiz questions. Each correct answer is worth +4 points and each wrong answer is -1 point. The final score was 135 points. We must find how many questions were wrong.

Givens

Total questions answered: 50
Right answer: +4 points
Wrong answer: -1 point
Final score: 135 points

Unknowns

The number of questions Ben got wrong

Constraints

Right count plus wrong count equals 50
Both counts are whole numbers from 0 to 50

2 · Planchoose the strategy

#6 Guess and Check · also uses: #9 Solve an Easier Related Problem#11 Work Backwards

Start from the easier 'all correct' case to get a clean total, then see how each wrong answer changes the score by a fixed amount. The fixed drop per wrong answer lets us divide to find the count.

3 · Execute4 carry out the plan

1Score if every answer were right

#9 Solve an Easier Related Problem 3.OA.A.3

If all 50 answers were correct, Ben would earn 4 points each.

50 \times 4 = 200

Pretending everything is right gives a simple multiplication to anchor the count.

2Find the points lost per wrong answer

#6 Guess and Check 3.OA.B.5

Turning one right answer into a wrong one loses the +4 it would have earned and adds a -1 penalty, so the score drops by 5 for each wrong answer.

4 + 1 = 5

Each mistake costs the missed 4 points plus a 1-point penalty, a steady 5 each time.

3Find how many points were lost in all

#11 Work Backwards 3.OA.D.8

The actual score is 135, which is 65 below the perfect 200.

200 - 135 = 65

The gap from a perfect score is the total damage all the wrong answers caused.

4Divide to find the wrong count

#6 Guess and Check 3.OA.A.3

Each wrong answer costs 5 points, and the total loss is 65, so divide to get the number of wrong answers.

65 \div 5 = 13

Sharing the 65 lost points into groups of 5 tells how many mistakes there were.

Answer: 13 questions

4 · Reviewdoes it hold up?

If 13 are wrong, 37 are right: 37 times 4 = 148, minus 13 = 135, which matches the final score exactly. The counts 37 and 13 add to 50.

Another way: Convert to a single-unknown equation (tool 13): with w wrong, right is 50 - w, so 4 times (50 - w) minus w = 135, giving 200 - 5w = 135 and w = 13.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the perfect score and dividing the lost points into groups of 5
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Reasoning that each wrong answer changes the score by a fixed 5 points
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the perfect score and the score gap across multiple steps

💡Takeaway. Pretend every answer was right, then count how far the score fell; each mistake costs the same amount, so just divide!

Variant 6 medium

In a math quiz contest, you earn $4$ points for each question you get right and lose $2$ point for each question you get wrong. If Nia solved $45$ questions in this contest and ended up with $138$ points, how many questions did Nia get wrong?

Your answer

Show solution

1 · Understandwhat's really being asked

Nia answered 45 quiz questions. Each correct answer is worth +4 points and each wrong answer is -2 point. The final score was 138 points. We must find how many questions were wrong.

Givens

Total questions answered: 45
Right answer: +4 points
Wrong answer: -2 point
Final score: 138 points

Unknowns

The number of questions Nia got wrong

Constraints

Right count plus wrong count equals 45
Both counts are whole numbers from 0 to 45

2 · Planchoose the strategy

#6 Guess and Check · also uses: #9 Solve an Easier Related Problem#11 Work Backwards

Start from the easier 'all correct' case to get a clean total, then see how each wrong answer changes the score by a fixed amount. The fixed drop per wrong answer lets us divide to find the count.

3 · Execute4 carry out the plan

1Score if every answer were right

#9 Solve an Easier Related Problem 3.OA.A.3

If all 45 answers were correct, Nia would earn 4 points each.

45 \times 4 = 180

Pretending everything is right gives a simple multiplication to anchor the count.

2Find the points lost per wrong answer

#6 Guess and Check 3.OA.B.5

Turning one right answer into a wrong one loses the +4 it would have earned and adds a -2 penalty, so the score drops by 6 for each wrong answer.

4 + 2 = 6

Each mistake costs the missed 4 points plus a 2-point penalty, a steady 6 each time.

3Find how many points were lost in all

#11 Work Backwards 3.OA.D.8

The actual score is 138, which is 42 below the perfect 180.

180 - 138 = 42

The gap from a perfect score is the total damage all the wrong answers caused.

4Divide to find the wrong count

#6 Guess and Check 3.OA.A.3

Each wrong answer costs 6 points, and the total loss is 42, so divide to get the number of wrong answers.

42 \div 6 = 7

Sharing the 42 lost points into groups of 6 tells how many mistakes there were.

Answer: 7 questions

4 · Reviewdoes it hold up?

If 7 are wrong, 38 are right: 38 times 4 = 152, minus 7 = 138, which matches the final score exactly. The counts 38 and 7 add to 45.

Another way: Convert to a single-unknown equation (tool 13): with w wrong, right is 45 - w, so 4 times (45 - w) minus w = 138, giving 180 - 6w = 138 and w = 7.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the perfect score and dividing the lost points into groups of 6
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Reasoning that each wrong answer changes the score by a fixed 6 points
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the perfect score and the score gap across multiple steps

💡Takeaway. Pretend every answer was right, then count how far the score fell; each mistake costs the same amount, so just divide!

Variant 7 medium

In a math quiz contest, you earn $4$ points for each question you get right and lose $1$ point for each question you get wrong. If Ava solved $50$ questions in this contest and ended up with $155$ points, how many questions did Ava get wrong?

Your answer

Show solution

1 · Understandwhat's really being asked

Ava answered 50 quiz questions. Each correct answer is worth +4 points and each wrong answer is -1 point. The final score was 155 points. We must find how many questions were wrong.

Givens

Total questions answered: 50
Right answer: +4 points
Wrong answer: -1 point
Final score: 155 points

Unknowns

The number of questions Ava got wrong

Constraints

Right count plus wrong count equals 50
Both counts are whole numbers from 0 to 50

2 · Planchoose the strategy

#6 Guess and Check · also uses: #9 Solve an Easier Related Problem#11 Work Backwards

Start from the easier 'all correct' case to get a clean total, then see how each wrong answer changes the score by a fixed amount. The fixed drop per wrong answer lets us divide to find the count.

3 · Execute4 carry out the plan

1Score if every answer were right

#9 Solve an Easier Related Problem 3.OA.A.3

If all 50 answers were correct, Ava would earn 4 points each.

50 \times 4 = 200

Pretending everything is right gives a simple multiplication to anchor the count.

2Find the points lost per wrong answer

#6 Guess and Check 3.OA.B.5

Turning one right answer into a wrong one loses the +4 it would have earned and adds a -1 penalty, so the score drops by 5 for each wrong answer.

4 + 1 = 5

Each mistake costs the missed 4 points plus a 1-point penalty, a steady 5 each time.

3Find how many points were lost in all

#11 Work Backwards 3.OA.D.8

The actual score is 155, which is 45 below the perfect 200.

200 - 155 = 45

The gap from a perfect score is the total damage all the wrong answers caused.

4Divide to find the wrong count

#6 Guess and Check 3.OA.A.3

Each wrong answer costs 5 points, and the total loss is 45, so divide to get the number of wrong answers.

45 \div 5 = 9

Sharing the 45 lost points into groups of 5 tells how many mistakes there were.

Answer: 9 questions

4 · Reviewdoes it hold up?

If 9 are wrong, 41 are right: 41 times 4 = 164, minus 9 = 155, which matches the final score exactly. The counts 41 and 9 add to 50.

Another way: Convert to a single-unknown equation (tool 13): with w wrong, right is 50 - w, so 4 times (50 - w) minus w = 155, giving 200 - 5w = 155 and w = 9.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the perfect score and dividing the lost points into groups of 5
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Reasoning that each wrong answer changes the score by a fixed 5 points
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the perfect score and the score gap across multiple steps

💡Takeaway. Pretend every answer was right, then count how far the score fell; each mistake costs the same amount, so just divide!

Variant 8 hard

In a math quiz contest, you earn $5$ points for each question you get right and lose $1$ point for each question you get wrong. If Mia solved $40$ questions in this contest and ended up with $158$ points, how many questions did Mia get wrong?

Your answer

Show solution

1 · Understandwhat's really being asked

Mia answered 40 quiz questions. Each correct answer is worth +5 points and each wrong answer is -1 point. The final score was 158 points. We must find how many questions were wrong.

Givens

Total questions answered: 40
Right answer: +5 points
Wrong answer: -1 point
Final score: 158 points

Unknowns

The number of questions Mia got wrong

Constraints

Right count plus wrong count equals 40
Both counts are whole numbers from 0 to 40

2 · Planchoose the strategy

#6 Guess and Check · also uses: #9 Solve an Easier Related Problem#11 Work Backwards

Start from the easier 'all correct' case to get a clean total, then see how each wrong answer changes the score by a fixed amount. The fixed drop per wrong answer lets us divide to find the count.

3 · Execute4 carry out the plan

1Score if every answer were right

#9 Solve an Easier Related Problem 3.OA.A.3

If all 40 answers were correct, Mia would earn 5 points each.

40 \times 5 = 200

Pretending everything is right gives a simple multiplication to anchor the count.

2Find the points lost per wrong answer

#6 Guess and Check 3.OA.B.5

Turning one right answer into a wrong one loses the +5 it would have earned and adds a -1 penalty, so the score drops by 6 for each wrong answer.

5 + 1 = 6

Each mistake costs the missed 5 points plus a 1-point penalty, a steady 6 each time.

3Find how many points were lost in all

#11 Work Backwards 3.OA.D.8

The actual score is 158, which is 42 below the perfect 200.

200 - 158 = 42

The gap from a perfect score is the total damage all the wrong answers caused.

4Divide to find the wrong count

#6 Guess and Check 3.OA.A.3

Each wrong answer costs 6 points, and the total loss is 42, so divide to get the number of wrong answers.

42 \div 6 = 7

Sharing the 42 lost points into groups of 6 tells how many mistakes there were.

Answer: 7 questions

4 · Reviewdoes it hold up?

If 7 are wrong, 33 are right: 33 times 5 = 165, minus 7 = 158, which matches the final score exactly. The counts 33 and 7 add to 40.

Another way: Convert to a single-unknown equation (tool 13): with w wrong, right is 40 - w, so 5 times (40 - w) minus w = 158, giving 200 - 6w = 158 and w = 7.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the perfect score and dividing the lost points into groups of 6
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Reasoning that each wrong answer changes the score by a fixed 6 points
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the perfect score and the score gap across multiple steps

💡Takeaway. Pretend every answer was right, then count how far the score fell; each mistake costs the same amount, so just divide!

Variant 9 hard

In a math quiz contest, you earn $2$ points for each question you get right and lose $1$ point for each question you get wrong. If Sam solved $100$ questions in this contest and ended up with $170$ points, how many questions did Sam get wrong?

Your answer

Show solution

1 · Understandwhat's really being asked

Sam answered 100 quiz questions. Each correct answer is worth +2 points and each wrong answer is -1 point. The final score was 170 points. We must find how many questions were wrong.

Givens

Total questions answered: 100
Right answer: +2 points
Wrong answer: -1 point
Final score: 170 points

Unknowns

The number of questions Sam got wrong

Constraints

Right count plus wrong count equals 100
Both counts are whole numbers from 0 to 100

2 · Planchoose the strategy

#6 Guess and Check · also uses: #9 Solve an Easier Related Problem#11 Work Backwards

Start from the easier 'all correct' case to get a clean total, then see how each wrong answer changes the score by a fixed amount. The fixed drop per wrong answer lets us divide to find the count.

3 · Execute4 carry out the plan

1Score if every answer were right

#9 Solve an Easier Related Problem 3.OA.A.3

If all 100 answers were correct, Sam would earn 2 points each.

100 \times 2 = 200

Pretending everything is right gives a simple multiplication to anchor the count.

2Find the points lost per wrong answer

#6 Guess and Check 3.OA.B.5

Turning one right answer into a wrong one loses the +2 it would have earned and adds a -1 penalty, so the score drops by 3 for each wrong answer.

2 + 1 = 3

Each mistake costs the missed 2 points plus a 1-point penalty, a steady 3 each time.

3Find how many points were lost in all

#11 Work Backwards 3.OA.D.8

The actual score is 170, which is 30 below the perfect 200.

200 - 170 = 30

The gap from a perfect score is the total damage all the wrong answers caused.

4Divide to find the wrong count

#6 Guess and Check 3.OA.A.3

Each wrong answer costs 3 points, and the total loss is 30, so divide to get the number of wrong answers.

30 \div 3 = 10

Sharing the 30 lost points into groups of 3 tells how many mistakes there were.

Answer: 10 questions

4 · Reviewdoes it hold up?

If 10 are wrong, 90 are right: 90 times 2 = 180, minus 10 = 170, which matches the final score exactly. The counts 90 and 10 add to 100.

Another way: Convert to a single-unknown equation (tool 13): with w wrong, right is 100 - w, so 2 times (100 - w) minus w = 170, giving 200 - 3w = 170 and w = 10.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the perfect score and dividing the lost points into groups of 3
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Reasoning that each wrong answer changes the score by a fixed 3 points
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the perfect score and the score gap across multiple steps

💡Takeaway. Pretend every answer was right, then count how far the score fell; each mistake costs the same amount, so just divide!

Variant 10 hard

In a math quiz contest, you earn $5$ points for each question you get right and lose $1$ point for each question you get wrong. If Ian solved $60$ questions in this contest and ended up with $246$ points, how many questions did Ian get wrong?

Your answer

Show solution

1 · Understandwhat's really being asked

Ian answered 60 quiz questions. Each correct answer is worth +5 points and each wrong answer is -1 point. The final score was 246 points. We must find how many questions were wrong.

Givens

Total questions answered: 60
Right answer: +5 points
Wrong answer: -1 point
Final score: 246 points

Unknowns

The number of questions Ian got wrong

Constraints

Right count plus wrong count equals 60
Both counts are whole numbers from 0 to 60

2 · Planchoose the strategy

#6 Guess and Check · also uses: #9 Solve an Easier Related Problem#11 Work Backwards

Start from the easier 'all correct' case to get a clean total, then see how each wrong answer changes the score by a fixed amount. The fixed drop per wrong answer lets us divide to find the count.

3 · Execute4 carry out the plan

1Score if every answer were right

#9 Solve an Easier Related Problem 3.OA.A.3

If all 60 answers were correct, Ian would earn 5 points each.

60 \times 5 = 300

Pretending everything is right gives a simple multiplication to anchor the count.

2Find the points lost per wrong answer

#6 Guess and Check 3.OA.B.5

Turning one right answer into a wrong one loses the +5 it would have earned and adds a -1 penalty, so the score drops by 6 for each wrong answer.

5 + 1 = 6

Each mistake costs the missed 5 points plus a 1-point penalty, a steady 6 each time.

3Find how many points were lost in all

#11 Work Backwards 3.OA.D.8

The actual score is 246, which is 54 below the perfect 300.

300 - 246 = 54

The gap from a perfect score is the total damage all the wrong answers caused.

4Divide to find the wrong count

#6 Guess and Check 3.OA.A.3

Each wrong answer costs 6 points, and the total loss is 54, so divide to get the number of wrong answers.

54 \div 6 = 9

Sharing the 54 lost points into groups of 6 tells how many mistakes there were.

Answer: 9 questions

4 · Reviewdoes it hold up?

If 9 are wrong, 51 are right: 51 times 5 = 255, minus 9 = 246, which matches the final score exactly. The counts 51 and 9 add to 60.

Another way: Convert to a single-unknown equation (tool 13): with w wrong, right is 60 - w, so 5 times (60 - w) minus w = 246, giving 300 - 6w = 246 and w = 9.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the perfect score and dividing the lost points into groups of 6
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Reasoning that each wrong answer changes the score by a fixed 6 points
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the perfect score and the score gap across multiple steps

💡Takeaway. Pretend every answer was right, then count how far the score fell; each mistake costs the same amount, so just divide!

Where next?

← Back to the problem: Find right and wrong counts from total score